Miguel Martinez

Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process

Abstract. In this article we extend the exact simulation methods of Beskos, Papaspiliopoulos and Roberts [Bernoulli 12 (2006), 1077–1098] to the solutions of one-dimensional stochastic differential equations involving the local time of the unknown process at point zero. In order to perform the method we compute the law of the skew Brownian motion with drift. The method presented in this article covers the case where the solution of the SDE with local time corresponds to a divergence form operator with a discontinuous coefficient at zero. Numerical examples are shown to illustrate the method and the performances are compared with more traditional discretization schemes.

Algebraic invariance conditions in the study of approximate (null-)controllability of Markov switch processes

We aim at studying approximate null-controllability properties of a particular class of piecewise linear Markov processes (Markovian switch systems). The criteria are given in terms of algebraic invariance and are easily computable. We propose several necessary conditions and a sufficient one. The hierarchy between these conditions is studied via suitable counterexamples. Equivalence criteria are given in abstract form for general dynamics and algebraic form for systems with constant coefficients or continuous switching. The problem is motivated by the study of lysis phenomena in biological organisms and price prediction on spike-driven commodities.

Approximations of a Continuous Time Filter. Application to Optimal Allocation Problems in Finance

Abstract In this article, we study a continuous time optimal filter and its various numerical approximations. This filter arises in an optimal allocation problem in the particular context of a non-stationary economy. We analyse the rates of convergence of the approximations of the filter when the model is misspecified and when the observations can only be made at discrete times. We give bounds that are uniform in time. Numerical results are presented.

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton---Jacobi Integrodifferential Systems on Networks

We study optimal control problems in infinite horizon whxen the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in Soner (SIAM J Control Optim 24(6):1110–1122, 1986) to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov’s “shaking the coefficients” method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton–Jacobi integrodifferential system. This ensures that the value function satisfies Perron’s preconization for the (unique) candidate to viscosity solution.